On the Cauchy problem for a linear harmonic oscillator with pure delay

Khusainov, Denys Ya; Pokojovy, Michael; Azizbayov, Elvin I.

On the Cauchy problem for a linear harmonic oscillator with pure delay

Dateien

Khusainov_0-296855.pdfGröße: 1.6 MBDownloads: 212

Datum

2015

Autor:innen

Khusainov, Denys Ya

Pokojovy, Michael

Azizbayov, Elvin I.

Open Access-Veröffentlichung

Open Access Gold

Sammlungen

Mathematik und Statistik

Publikationstyp

Zeitschriftenartikel

Publikationsstatus

Published

Erschienen in

Advances in Difference Equations. 2015, 2015, 197. ISSN 1687-1839. eISSN 1687-1847. Available under: doi: 10.1186/s13662-015-0538-z

Zusammenfassung

In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace so that the solution of the homogeneous problem can be written as a linear combination of these two eigenfunctions. As opposed to that, in the presence even of a small delay, the spectrum is infinite and a finite sum representation is not possible. Using a special function referred to as the delay exponential function, we give an explicit solution representation for the Cauchy problem associated with the linear oscillator with pure delay. Finally, the solution asymptotics as the delay parameter goes to zero is studied. In contrast to earlier works, no positivity conditions are imposed.

Fachgebiet (DDC)

510 Mathematik

Schlagwörter

functional-differential equations, harmonic oscillator, pure delay, well-posedness, solution representation

Zitieren

ISO 690

KHUSAINOV, Denys Ya, Michael POKOJOVY, Elvin I. AZIZBAYOV, 2015. On the Cauchy problem for a linear harmonic oscillator with pure delay. In: Advances in Difference Equations. 2015, 2015, 197. ISSN 1687-1839. eISSN 1687-1847. Available under: doi: 10.1186/s13662-015-0538-z

BibTex

@article{Khusainov2015Cauch-32063,
  year={2015},
  doi={10.1186/s13662-015-0538-z},
  title={On the Cauchy problem for a linear harmonic oscillator with pure delay},
  volume={2015},
  issn={1687-1839},
  journal={Advances in Difference Equations},
  author={Khusainov, Denys Ya and Pokojovy, Michael and Azizbayov, Elvin I.},
  note={Article Number: 197}
}

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